Let $g$ be a differentiable function with $g(4)=6$ and $g'(4)=-2$. What is the value of the approximation of $g(4.2)$ using the function's local linear approximation at $x=4$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $5.3$ (Choice B) B $5.4$ (Choice C) C $5.5$ (Choice D) D $5.6$
The local linear approximation of $g$ at $x=4$ is achieved using the equation of the line tangent to $g$ at $x=4$. Let $L(x)$ represent this equation. We can find $L(x)$ using the general formula for the tangent to the graph of function $u$ at $x=a$ : $y=u'(a)(x-a)+u(a)$ [Is there a way to find this formula without memorizing?] In our case, $L(x)=g'(4)(x-4)+g(4)$. Plugging $g(4)=6$ and $g'(4)=-2$, we obtain $L(x)=-2(x-4)+6$. To approximate $g(4.2)$, all we need is to plug $x=4.2$ into $L(x)$. $\begin{aligned} L(4.2)&=-2(4.2-4)+6 \\\\ &=-2(0.2)+6 \\\\ &=5.6 \end{aligned}$ In conclusion, the approximation of $g(4.2)$ using the function's local linear approximation at $x=4$ is $5.6$.